Numerical simulations of the propagation of Chirped Vector Soliton in optical fibers with variable
coefficients in the presence of third order dispersion and power law nonlinearity
Aziez Siham1,2
1Research Center in Industrial Technologies CRTI P. O. Box 64, Cheraga 16014, Algiers
aziez_siham@hotmail.fr
Bahloul Derradji2
2Département de physique, Faculté des Sciences, Université Hadj-Lakhdar de Batna
1 Avenue Boukhlouf Mohamed El Hadi, Batna 05000, Algeria
Abstract— We study in this work, the numerical simulations of propagation of chirped vector solitons in optical fiber systems using the compact split step Padé scheme (CSSPS). This study is done in the case of variable coefficients and in the presence of third order dispersion and power law nonlinearity. A negative chirp makes the soliton broadening, while; a positive chirp leads to a soliton compression. The effect of chirp on the soliton temporal width of an amplification system is greater than that in a loss system. In the presence of third order dispersion, we note an increase of the pulse width with an asymmetric oscillation on the trailing edge. In the same time, we note a shift of the center of the two components of the one managed chirped vector soliton along the propagation distance. It is clearly noted from plot that, the quintic nonlinearity has a marginal role on the propagation characteristics of the two components of managed chirped vector soliton.
Keywords— Vector solitons, chirped solitons, birefringent optical fibers, compact split step Padé scheme, and coupled higher- order nonlinear Schrodinger equations with variable coefficients
Introduction
Since its first observation and their propagation in optical fibers [1,2,3], optical solitons [1,2,3] have a lot of interest because of their possibility to propagate in the anomalous dispersion region of optical fibers on considerable distances without broadening, due to the balance between group velocity dispersion and nonlinear self-phase modulation. This balance in single mode fibers can be described by the nonlinear Schrödinger equation [1,2,3].
In reality, due to the presence of the birefringence [1,2,4,6,8], the real single mode fibers are not truly single modal but bimodal. The propagation of optical solitons in birefringent optical fibers can be described by a system of coupled nonlinear Schrödinger (CNLS) equations, which can be derived in various physical contexts [1,2]. In this case, the CNLS equations may give as a solution a vector soliton.
Optical vector soliton illustrate the most attention due to their
wide range of applications in fiber-optic-based communication systems [9], and their multicomponent structure [4].
The propagation of managed vector solitons in birefringent optical fibers is governed by the coupled NLS equations [1,2,4,6,8] with variable coefficients. The coupled NLS equations [1,2,4,6,8] with variable coefficients are not integrable except in some particular cases. So, to study the propagation of managed vector solitons, it is necessary to use numerical methods. In this work, we use the compact split step Padé scheme (CSSPS) [5]. CSSPS method is more efficient, more rapid and well adapted for higher order time derivatives such as third order dispersion or Kerr dispersion.
I. THEORETICAL MODEL
In real systems, it is found that the dispersion, nonlinearity, gain and loss are usually varied with the propagation distance.
In this case, the propagation of managed vector solitons is governed by the CNLS equation [1,2,4,6,8] with variable coefficients:
4 0 4
4 0 4
2 0 2
2 0 2
3 3 6 1 2 2 2
1 0
0 1
3
v u v i u
v u v Z u
iR
v u
ZZ T T
iD
v u v T
u Z
(1)
Whereu(Z,T)andv(Z,T)are the slowly varying amplitudes of the two polarization modes in the fiber. Z and T are
0 2
4 6
8 10 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
dis tanc e z /LD Intens ity Ix
tim e t/T0
Ix(Z,T)
0 2
4 6
8 10 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
dis tanc e z /LD Intens ity Iy
tim e t/T0
Iy(Z,T)
0 2
4 6
8 10 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
dis tanc e z /LD Intens ity Ix
tim e t/T0
Ix(Z,T)
0 2
4 6
8 10 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
distance z/LD Intensity Iy
tim e t/T0
Iy(Z,T)
respectively the normalized distance and time. δ is the difference of the group velocities between the two polarization components. β is the cross-phase modulation (XPM) coefficient (For linearly birefringent fibers, β=2/3). K is the quintic nonlinearity coefficient.β3is the third order dispersion (TOD). The functions D(Z), R(Z), Г(Z) are respectively the managed group velocity dispersion (GVD), managed self- phase modulation (SPM) and is the linear and nonlinear gain (loss).
The parameters D(Z), R(Z) and Г(Z) are given by the following expressions[7]:
the varying GVD parameter
Z exp
Z RZ /D0D (2)
the nonlinear parameter
Z R R sin
gZR 0 1 (3)
and the gain or loss distributed parameter
2 Z (4)
WhereD0is a parameter related to the initial peak power,σ is a parameter describing gain or loss, and R0, R1, g are the parameters describing Kerr nonlinearity. For our study, we take the parametersD0=1,R0=0,R1=1andg=1,σ=0[7].
II. NUMERICAL MODEL
In order to do our numerical simulations, we choose the following initial conditions with linear chirp:
,T cos exp iCT /cosh
Tu
2
0 2 (5)
,T sin exp iCT /cosh
Tv
2
0 2 (6)
Where C is the linear chirp parameter and the angle α determines the relative strength of the partial pulses in each of the two polarizations. It represents the angle of the polarization of the soliton with respect to the polarization axis of the fiber.
III. NUMERICAL SIMULATIONS
In the following, we examine the evolution of managed vector soliton under the effect of the chirp, the quintic nonlinearity, and the third order dispersion when α=45° (the two polarization components have the same amplitude).
Starting our discussions by the simple case where we will examine the evolution of managed vector soliton under the effect of the chirp whereβ3andKare set to zero. The profile of the intensity of the two components of polarization of chirped vector soliton are depicted in Fig. 1. (negative chirp C<0) and Fig. 2. (positive chirpC>0). A negative chirp leads to the optical vector soliton broadening, while; a positive chirp leads to a vector soliton compression with the increase of propagation distance.
Fig. 1. Evolution of the two components of one managed chirped vector soliton in birefringent optical fiber with the parameters α=45°,C=-0.2and σ=0.
Fig. 2. Evolution of the two components of one managed chirped vector soliton in birefringent optical fiber with the parameters α=45°,C=0.2 and σ=0.
0 5
10
-20 -15 -10 -5 0 5 10 15 20
0 0.1 0.2 0.3 0.4 0.5
distance z/LD Intensity Ix
time t/T0
Ix(Z,T)
0 5
10
-20 -15 -10 -5 0 5 10 15 20
0 0.1 0.2 0.3 0.4 0.5
dis tanc e z /LD Intens ity Iy
tim e t/T0
Iy(Z,T)
0 2
4 6
8 10 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
distance z/LD Intensity Ix
tim e t/T0
Ix(Z,T)
0 2
4 6
8 10 -10
-5 0
5 10 0
0.2 0.4 0.6 0.8
dis tanc e z /LD Intens ity Iy
tim e t/T0
Iy(Z,T)
Fig. 3. Evolution of the two components of one managed chirped vector soliton in birefringent optical fiber with the parametersα=45°,C=0.2,β3=1, β4=0,K=0andσ=0.
Fig. 4. Evolution of the two components of one managed chirped vector soliton in birefringent optical fiber with the parametersα=45°,C=0.2,β3=0, β4=0,K= -0.05andσ=0.
To study the effect of TOD alone, the quintic nonlinearity Kis set to zero in (1). Fig. 3. shows the evolution of the two components of the managed chirped vector soliton for the following parameters α=45°, C=0.2, β3=1, β4=0, K=0, and σ=0. We note that the pulse width of the two components of the one managed chirped vector soliton increases along the propagation distance. At the same time, there is an asymmetric oscillation on the trailing edge and a shift of the center of the two components of the managed chirped vector soliton.
From Fig. 4., it is clear that the role of quintic nonlinearity on the propagation characteristics of the two components of the managed chirped vector soliton is unimportant.
Conclusion
In this paper, the propagation characteristics of chirped managed vector solitons in birefringent optical fibers is studied numerically in the presence of third order dispersion and quintic nonlinearity using the compact split step Padé scheme (CSSPS). The propagation of vector solitons has a rich nonlinear dynamics. A negative chirp makes the chirped managed vector soliton broadening, while; a positive chirp leads to a chirped managed vector soliton compression. In the presence of third order dispersion, we note an increase of the pulse width with an asymmetric oscillation on the trailing edge and a shift of the center of the two components of the managed chirped vector soliton along the propagation distance.. From plot, it is clearly noted that, the quintic nonlinearity has a marginal role on the propagation characteristics of the two components of the managed chirped vector soliton.
Acknowledgment
This work is supported by the post graduation of
« Research Center in Industrial Technologies CRTI - P. O.
Box 64, Cheraga 16014, Algiers », and the “Département de Science, Faculté des Sciences de l’ Université de Batna, Algeria”.
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